# Lesson 12

Proofs about Quadrilaterals

- Let’s prove theorems about quadrilaterals and their diagonals.

### Problem 1

Lin is using the diagram to prove the statement, “If a parallelogram has one right angle, it is a rectangle.” Given that \(EFGH\) is a parallelogram and angle \(HEF\) is right, which reasoning about angles will help her prove that angle \(FGH\) is also a right angle?

Corresponding angles are congruent when parallel lines are cut by a transversal.

Opposite angles in a parallelogram are congruent.

Vertical angles are congruent.

The base angles of an isosceles triangle are congruent.

### Problem 2

\(ABDE\) is an isosceles trapezoid. Select **all** pairs of congruent triangles.

Triangle \(ABE\) and triangle \(DBE\)

Triangle \(ABD\) and triangle \(DAE\)

Triangle \(ABE\) and triangle \(BAD\)

Triangle \(AED\) and triangle \(BDE\)

Triangle \(EAB\) and triangle \(EDB\)

### Problem 3

Match each conjecture with the rephrased statement of proof connected to the diagram.

### Problem 4

Which of the following criteria *always* proves triangles congruent? Select **all** that apply.

Corresponding congruent Angle-Side-Angle

Corresponding congruent Side-Angle-Side

Corresponding congruent Side-Side-Angle

3 congruent sides

2 congruent sides

3 congruent angles

### Problem 5

Select **all** true statements based on the diagram.

Segment \(EB\) is congruent to segment \(AD\).

Segment \(DC\) is congruent to segment \(AB\).

Segment \(DA\) is congruent to segment \(CB\).

Angle \(CBE\) is congruent to angle \(ABE\).

Angle \(CEB\) is congruent to angle \(DEA\).

Line \(DA\) is parallel to line \(CB\).

Line \(DC\) is parallel to line \(AB\).

### Problem 6

Diego states that diagonal \(WY\) bisects angles \(ZWX\) and \(ZYX\). Is he correct? Explain your reasoning.

### Problem 7

Sketch the unique triangles that can be made with angle measures \(80^{\circ}\) and \(20^{\circ}\) and side length 5. How do you know you have sketched all possibilities?